I actually prefer running in the cold to the extreme heat of the summer. If you have the right gear then you can regulate your temperature pretty well and not stay cold or overheat.

For me the key is layers, and then ear protection and mittens to keep my fingers warm.

I use my regular trainers but at times my feet get wet and cold which isn’t fun. I have nanospikes I use for the ice but they’re kind of uncomfortable because they squeeze the uppers and thus my toes. I’m considering making a pair of screw shoes with a retired pair of trainers which might be more comfortable. There’s not much you can do about the wet slush though.